Question

Country A and country B both have the production function • Y 5 F(K, L ) 5 K 1/3 2/3 L .

What happens subsequently to output per worker in the postwar economy? Is the growth rate of output per worker after the war smaller or greater than it was before the war?

a. Does this production function have constant returns to scale? Explain.

b. What is the per-worker production function, y 5 f (k)?

Answer 1

Answer:

a) Yes

$Y= (\lambda K)^{1/3} (\lambda L)^{2/3}$

$Y= \lambda^{1} K^{1/3]L^{2/3}$

$Y= a y =\lambda x$

b) $\frac{Y}{L} =\frac{K^{1/3}L^{2/3}}{L}$

$\frac{Y}{L}=K^{1/3}L^{-1/3}$

And we can express like this:

$Y= (\frac{K}{L})^{1/3}$

$Y= a^{1/3}$ where a is a cosntant. And that would represent the per-worker production function.

When we have a war, we observe a reduction in the labor force and that is translate to math that L decrease, and as a consecuence the Capital-labor ratio k = K/L increase ,since L decrease. So then we can conclude that production output decrease  since we have less workers, by the way the output procuded per each worker increase, and as we have that each worker  have more capital.  

Explanation:

For this case we have the following production function for Country A and B:

$Y=F(K,L) = K^{1/3}L^{2/3}$

Part a, in order to see if we have constant returns to scale we need to show that the function Y we have a linear form Y= aX, wher a is the slope.

If the production function has constant, increasing or decreasing returns to scale w can detemrined this multiplying the capital and labor by a constant $\lambda$

$Y= (\lambda K)^{1/3} (\lambda L)^{2/3}$

Doing algebra we got:

$Y= \lambda^{1} K^{1/3]L^{2/3}$

And we can express the last expression like this:

$Y= a y =\lambda x$

And we can conclude that whn the capital and labour increasse by a factor of a the output increase by a too.

So then the answer for this case is yes.

Part b

In order to find the per workr production function we need to divide out function by L and if we do this we got that:

$\frac{Y}{L} =\frac{K^{1/3}L^{2/3}}{L}$

$\frac{Y}{L}=K^{1/3}L^{-1/3}$

And we can express like this:

$Y= (\frac{K}{L})^{1/3}$

And can be expressed like this:

$Y= a^{1/3}$ where a is a cosntant. And that would represent the per-worker production function.

When we have a war, we observe a reduction in the labor force and that is translate to math that L decrease, and as a consecuence the Capital-labor ratio k = K/L increase ,since L decrease. So then we can conclude that production output decrease  since we have less workers, by the way the output procuded per each worker increase, and as we have that each worker  have more capital.